Acta Mathematica Sinica, English Series

, Volume 34, Issue 8, pp 1195–1207 | Cite as

Minimal Complex Surfaces with Levi–Civita Ricci-flat Metrics

  • Ke Feng LiuEmail author
  • Xiao Kui Yang


This is a continuation of our previous paper [14]. In [14], we introduced the first Aeppli–Chern class on compact complex manifolds, and proved that the (1, 1) curvature form of the Levi–Civita connection represents the first Aeppli–Chern class which is a natural link between Riemannian geometry and complex geometry. In this paper, we study the geometry of compact complex manifolds with Levi–Civita Ricci-flat metrics and classify minimal complex surfaces with Levi–Civita Ricci-flat metrics. More precisely, we show that minimal complex surfaces admitting Levi–Civita Ricci-flat metrics are Kähler Calabi–Yau surfaces and Hopf surfaces.


Levi–Civita Ricci-flat metric kodaira dimension classication 

MR(2010) Subject Classification

53C55 32Q25 32Q20 


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The second author would like to thank Valentino Tosatti for many useful comments and suggestions.


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Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsCapital Normal UniversityBeijingP. R. China
  2. 2.Department of MathematicsUniversity of California at Los AngelesCaliforniaUSA
  3. 3.Morningside Center of Mathematics, Institute of Mathematics, Hua Loo-Keng center of Mathematical Sciences, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingP. R. China

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